Operator Space Characterizations of C*-algebras and Ternary Rings

In the category of operator spaces, that is, subspaces of the bounded linear operators B(H) on a complex Hilbert space H together with the induced matricial operator norm structure, objects are equivalent if they are completely isometric, i.e., if there is a linear isomorphism between the spaces which preserves this matricial norm structure. Since operator algebras, that is, subalgebras of B(H), are motivating examples for much of operator space theory, it is natural to ask if one can characterize which operator spaces are operator algebras. One satisfying answer was given by Blecher, Ruan and Sinclair in [10], where it was shown that among operator spaces A with a (unital but not necessarily associative) Banach algebra product, those which are completely isometric to operator algebras are precisely the ones whose multiplication is completely contractive with respect to the Haagerup norm on A⊗A. (For a completely bounded version of this result, see [7].) A natural object to characterize in this context are the so called ternary rings of operators (TRO’s). These are subspaces of B(H) which are closed under the ternary product xy∗z. This class includes C*-algebras. TRO’s, like C*-algebras, carry a natural operator space structure. In fact, every TRO is (completely) isometric to a corner pA(1 − p) of a C*-algebra A. TRO’s are important because, as shown by Ruan [35], the injectives in the category of operator spaces are TRO’s (corners of injective C∗-algebras) and not, in general, operator algebras. (For the dual version of this result see [15].) Injective envelopes of operator systems and of operator spaces ([23] and [35]) have proven to be important tools, see for example [9]. The characterization of TRO’s among operator spaces is the subject of this paper. (See Theorem 5.3.) Closely related to TRO’s are the so called JC*-triples, norm closed subspaces of B(H) which are closed under the triple product (xy∗z + zy∗x)/2.